Find equations of two circles which are drawn through the origin to cut the circle $$x^2+y^2-x+3y-1=0$$ orthogonally and to touch the line $$x+2y+1=0$$.
$$x^2+y^2-2ax-2by=0$$----(1) is the general equation and $$b=a+1/3$$----(2) is the locus of mid points of those circles. I plugged $$y=(-x-1)/2$$(tangent) to $$y$$ of (1) and (2) to the $$b$$ of (1). Thus
$$15x^2+(10-20a)x+(4a+4)$$
I need some advice on where to go from here.
Two circles through origin touching the line $x+2y+1=0$ are fixed. They do not necessarily cut an arbitrarily positioned circle orthogonally.
Either graph/ rough sketch to convince yourself what possibility could exist for an orthogonal intersection.